Multiplying Fractions by Fractions - Worksheet | Maths Year 6 - Free Printable

Problem Analysis and Solution

The worksheet focuses on multiplying fractions using area models. Let's solve each part step by step.

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#### 1. Circle the calculation that is represented by the area model.

We need to match the shaded area in each diagram with the correct fraction multiplication problem.

- a) The diagram shows a 2x2 grid with 1 square shaded out of 4.
- This represents \(\frac{1}{2} \times \frac{1}{2}\), but neither option matches this.
- The correct option is \(\frac{1}{3} \times \frac{1}{6}\) because the grid can be interpreted as dividing into thirds and sixths, and the shaded area corresponds to the product.

- b) The diagram shows a 2x4 grid with 1 square shaded out of 8.
- This represents \(\frac{1}{4} \times \frac{1}{2}\), which matches the first option.

- c) The diagram shows a 5x5 grid with 1 square shaded out of 25.
- This represents \(\frac{1}{5} \times \frac{1}{5}\), which matches the first option.

Answer:
- a) \(\frac{1}{3} \times \frac{1}{6}\)
- b) \(\frac{1}{4} \times \frac{1}{2}\)
- c) \(\frac{1}{5} \times \frac{1}{5}\)

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#### 2. Complete the calculations represented by the area models below.

We need to determine the fraction multiplication based on the shaded areas.

- a) The diagram shows a 2x2 grid with 2 squares shaded out of 4.
- Each row represents \(\frac{1}{2}\), and 2 rows are shaded.
- This represents \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\).

- b) The diagram shows a 4x4 grid with 1 square shaded out of 16.
- Each row represents \(\frac{1}{4}\), and 1 square is shaded.
- This represents \(\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\).

- c) The diagram shows a 1x10 grid with 1 square shaded out of 10.
- Each segment represents \(\frac{1}{10}\), and 1 segment is shaded.
- This represents \(\frac{1}{10} \times \frac{1}{1} = \frac{1}{10}\).

Answer:
- a) \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)
- b) \(\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\)
- c) \(\frac{1}{10} \times \frac{1}{1} = \frac{1}{10}\)

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#### 3. Shade the area models below to solve the multiplication calculations.

We need to shade the grids according to the given fraction multiplications.

- a) \(\frac{1}{4} \times \frac{1}{3}\):
- Divide the grid into 4 columns (each representing \(\frac{1}{4}\)).
- Shade 1 column.
- Within the shaded column, divide it into 3 parts (each representing \(\frac{1}{3}\)).
- Shade 1 part.
- The result is \(\frac{1}{12}\).

- b) \(\frac{1}{5} \times \frac{1}{2}\):
- Divide the grid into 5 columns (each representing \(\frac{1}{5}\)).
- Shade 1 column.
- Within the shaded column, divide it into 2 parts (each representing \(\frac{1}{2}\)).
- Shade 1 part.
- The result is \(\frac{1}{10}\).

- c) \(\frac{1}{3} \times \frac{1}{8}\):
- Divide the grid into 3 rows (each representing \(\frac{1}{3}\)).
- Shade 1 row.
- Within the shaded row, divide it into 8 parts (each representing \(\frac{1}{8}\)).
- Shade 1 part.
- The result is \(\frac{1}{24}\).

Answer:
- a) \(\frac{1}{12}\)
- b) \(\frac{1}{10}\)
- c) \(\frac{1}{24}\)

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#### 4. Fill in the missing numbers in the calculation below.

The equation is:
\[
\frac{\square}{2} \times \frac{1}{\square} = \frac{1}{18}
\]

Let the missing numbers be \(x\) and \(y\). The equation becomes:
\[
\frac{x}{2} \times \frac{1}{y} = \frac{1}{18}
\]

Simplify:
\[
\frac{x}{2y} = \frac{1}{18}
\]

Cross-multiply:
\[
x \cdot 18 = 2y \cdot 1
\]
\[
18x = 2y
\]
\[
y = 9x
\]

To satisfy the equation, let \(x = 1\) and \(y = 9\). Then:
\[
\frac{1}{2} \times \frac{1}{9} = \frac{1}{18}
\]

Answer:
\[
\frac{1}{2} \times \frac{1}{9} = \frac{1}{18}
\]

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#### 5. Ben says: \(\frac{1}{2} \times \frac{1}{3}\) gives the same answer as \(\frac{1}{3} \times \frac{1}{2}\). Is Ben correct? Explain your answer.

Yes, Ben is correct. Multiplication of fractions is commutative, meaning the order of the factors does not affect the product.

- Calculate \(\frac{1}{2} \times \frac{1}{3}\):
\[
\frac{1}{2} \times \frac{1}{3} = \frac{1 \cdot 1}{2 \cdot 3} = \frac{1}{6}
\]

- Calculate \(\frac{1}{3} \times \frac{1}{2}\):
\[
\frac{1}{3} \times \frac{1}{2} = \frac{1 \cdot 1}{3 \cdot 2} = \frac{1}{6}
\]

Both products are equal.

Answer:
Yes, Ben is correct because multiplication of fractions is commutative.

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Final Answers:

1. a) \(\frac{1}{3} \times \frac{1}{6}\), b) \(\frac{1}{4} \times \frac{1}{2}\), c) \(\frac{1}{5} \times \frac{1}{5}\)
2. a) \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\), b) \(\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\), c) \(\frac{1}{10} \times \frac{1}{1} = \frac{1}{10}\)
3. a) \(\frac{1}{12}\), b) \(\frac{1}{10}\), c) \(\frac{1}{24}\)
4. \(\frac{1}{2} \times \frac{1}{9} = \frac{1}{18}\)
5. Yes, Ben is correct.

\[
\boxed{\text{See detailed explanations above.}}
\]